velopment and application of its principles; for lines, surfaces, or solids, can be expressed only in relation to their magnitudes or figures, and an expression of magnitude is the result of comparison with some other magnitude of the same kind as a unit.

DEFINITIONS OF TERMS USED IN GEOMETRY.

An Ariom is a self-evident truth.

A Postulate requires us to admit the possibility of an operation.

A Theorem is a truth which becomes evident by means of a train of reasoning called a demonstration

A Demonstration is direct, when the truth is inferred directly from the premises, as the conclusion of a regular series of inductions.

A Demonstration is indirect, when the conclusion shows that the introduction of any supposition, contrary to the truth advanced, necessarily leads to an absurdity.

A Problem proposes an operation to be performed.

A Lemma is a subsidiary truth, the evidence of which must be established, preparatory to the demonstration of a succeeding theorem.

A Proposition is a general term for either a theorem, a problem or a lemma.

A Corollary is an obvious consequence, resulting from a demonstration.

An Hypothesis is a supposition, and may be either true or false.

A Scholium is a remark on one or several preceding propositions, which tends to point out their connection, their use, their restriction, or their extension.

EXPLANATION OF SYMBOLS EMPLOYED.

1. The sign is the sign of equality between two or more magnitudes or quantities; thus, the expression A=B signifies that A is equal to B.

2. The symbol placed between two magnitudes or quantities denotes that the former is greater than the latter; the symbol is used to denote that the former is less than the latter. Thus, AB indicates that the quantity represented by A is greater than that represented by B: by A<B is implied that A is less than B.

3. The sign is called plus: it indicates addition.

4. The sign is called minus: it signifies subtraction.

Thus A+B represents the sum of the quantities A and B : A-B represents their difference, or what remains after B is taken from A; and A-B+C, or A+C-B, signifies that A and are to be added together, and that B is to be subtracted from their sum. The double sign is employed to express indiferently either the sum or difference of the magnitudes or quantities between which it is placed.

5. The sign is used to denote the difference between two quantities when it is not known which is the greater; thus AB represents the difference between A and B, without reference to which is the greater.

6. The sign X, into or with, indicates multiplication: thus AXB represents the product of A and B. A point is sometimes used instead of the sign X; thus A.B denotes the same as AXB. The same product is also designated without any intermediate sign, by AB. The expression AX(B +C-D) represents the product of A by the quantity B+C -D. If A+B were to be multiplied by A-B+C, the product would be indicated thus (A+B)×(A—B+C), those quantities enclosed within the brackets being considered as a single quantity.

7. A number placed before a letter denoting a quantity, is called a co-efficient, and serves as a multiplier to that quantity: thus 3A signifies that the quantity A is to be repeated three times: A signifies half the quantity A.

8. The sign siguifies division, and is called by. Thus A÷B signifies that the quantity A is to be divided by the quantity B; the same is also represented by placing the dividend above the divisor, thus is equivalent to the expression A÷B.

A

B

9. A power of a quantity is denoted by affixing to the quantity a small figure or letter indicating the degree of the power, the figure being a little elevated to distinguish it from a multiplier. Thus A signifies the square of A, or that A is to be multiplied into itself; A signifies the cube of A, or that the product of A into itself is to be multiplied by A; and A denotes that A is to be raised to the "th power, or multiplied into itself" number of times.

10. The sign indicates a root to be extracted; thus means the square root of 2; AXB or ✔(AXB) means the square root of the product of A and B.

11. The symbol.. is used to denote the word hence or therefore; indicating a conclusion, depending on the foregoing premises.

AXIOMS.

1. Quantities which are equal to the same quantities, are equal to each other.

2. If equals be added to equals, the wholes will be equal. 3. If equals be taken from equals, the remainders will be equal.

4. If equals be added to unequals, the wholes will be unequal.

5. If equals be taken from unequals, the remainders will be unequal.

6. If equal quantities are multiplied by equal quantities, the products will be equal.

7. If equal quantities are divided by equal quantities, the quotients will be equal.

8. The whole of a quantity is equal to the sum of all its parts.

9. Things which coincide or fill the same space are identical or mutually equal in all their parts.

POSTULATES.

1. Let it be granted that magnitudes, as well as numbers, may be multiplied indefinitely.

2. Let it be granted that magnitudes, as well as numbers, may be divided indefinitely.

3. Let it be granted that one quantity or magnitude may be added to another quantity or magnitude. Also, that any number of quantities or magnitudes may be added to any number of other quantities or magnitudes, of the same kind.

4. Let it also be granted that any quantity or magnitude may be taken from any other greater quantity or magnitude, of the same kind.

5. And hence, that any quantity or magnitude may be increased till it is equal to a greater, or diminished till it is equal to a less.

ON THE RELATION OF MAGNITUDES TO NUMBERS, AND ON RATIOS AND PROPORTIONS.

Definitions and Explanation of Principles.

1. Number is the expression of an object, or an association of objects, indicating their individuality, or the magnitude, of such association.

Magnitudes to be represented by numbers must be re

ferred to some specific magnitude of the same kind as a unit; and this magnitude so represented by a unit, may be called a prime; and any magnitude may be expressed by the number of such primes contained in such magnitude.

Magnitudes so represented by numbers may be compared with each other, and hence results ratio.

2. Ratio is a mutual relation of two magnitudes of the same kind to each other in respect of quantity. Or it is the quotient arising from dividing one quantity by another quantity of the same kind. Thus, if A and B represent quantities of the same kind, the ratio of A to B is expressed by Or the ratio is sometimes expressed by placing a colon between the magnitudes compared, thus the ratio of A to B is A: B.

A

B

Of two magnitudes A and B, if A be divided into M number of units, each equal to A', then A=MXA'; and if B be divided into N number of units, each equal to A', then B= NXA'; M and N being integral numbers. The ratio of A to B will be the same as the ratio of MXA' to NXA'; that is as the ratio of M to N, since A' is a common unit.

In the same manner the ratio of any other two magnitudes, C and D, may be expressed by PXC' to QXC', P and Q being also integral numbers, and their ratio will be the same as that of P to Q.

3. The two magnitudes, when spoken of separately, are called the terms of the ratio; when spoken of together, they are called a couplet; the first term is called the antecedent and the last term the consequent.

4. A compound ratio is the ratio of the products of the corresponding terms of two or more simple ratios: thus the ratio of A to B is and the ratio of C to D is ; and the

B

A'

B D

D

BD

ratio compounded of these two ratios is x or
C AC

5. That class of compound ratios produced by multiplying a simple ratio into itself or into another equal ratio, that is, the square of a simple ratio, is called a duplicate ratio; that produced by multiplying three equal ratios together, that is, the cube of a simple ratio, is called a triplicate ratio, &c.; the ratio of the square roots of two magnitudes is called a subduplicate ratio; that of the cube roots, a subtriplicate, &c. Thus the simple ratio of A to B is A: B.

The duplicate ratio is

A2: B2

A3: B

The subduplicate ratio is

NAB

The triplicate ratio is

The subtriplicate ratio is

A. B

6. To multiply the antecedent by any quantity, divides the ratio by that quantity; and to divide the antecedent, multiplies the ratio.

And conversely, to multiply the consequent of a couplet by any quantity, multiplies the ratio, and to divide the consequent, divides the ratio.

7. To multiply or divide both the antecedent and consequent by the same quantity does not alter the ratio.

B

A

Thus the ratio of A to B is

The ratio of AX2: BX2 is

The ratio of A÷2: B÷2 is

S. If four magnitudes A, B,

B

is equal to then A is said to have the same ratio to B

C'

that C has to D. When four quantities have this relation to each other, they are said to be in proportion.

Proportion, then, is an equality of ratios.

9. To indicate that the ratio of A to B is equal to the ratio of C to D, the quantities are usually written thus, A: B:: C: D, and read, A is to B as C to D; the quantities which are compared together are called the terms of the proportion; the first and last terms are called the two extremes, and the second and third terms the two means.

10. Of four proportional quantities the first and third are called the antecedents, and the second and fourth the consequents; and the last is said to be a fourth proportional to the other three taken in order. Antecedents are called homologous or like terms, and so of the consequents.

11. Magnitudes are said to be in continued proportion when every consequent is considered as the antecedent of the succeeding term; thus if A is to B as B to C and as C to D, those magnitudes are in continued proportion; and are written A: B::B: C::C: D, &c.

Several independent couplets having the same ratio are frequently expressed in the same way. Thus if A is to B as C to D and as E to F, they are written A: B::C: D::E: F.